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Abstract:

Illumination lenses having surfaces shaped according to given
differential equations in order to distribute light in a highly
controlled manner with minimum reflection losses are provided. These
lenses also have surfaces angled and arranged in such a way that they can
be molded with relatively simple molds.

8. A method of making a lens comprising: integrating at least one
differential equation selected from the group consisting of DE1, DE2,
DE3, DE4 and DE5 presented above to obtain at least one lens surface
generatrice; producing a mold that is shaped according to the at least
one lens surface generatrice; and producing lenses using the mold.

Description:

FIELD OF THE INVENTION

[0001] The present invention relates to illumination optics especially
suitable for use with Light Emitting Diodes.

BACKGROUND OF THE INVENTION

[0002] Traditionally Light Emitting Diodes (LEDs) have primarily been used
as indicator lamps in electronic equipment. However recently the power
and efficicacy (e.g., lumens per watt of electrical power) has been
increasing and LEDs have been identified as a possible replacement for
inefficient incandescent lamps in certain applications. The light
emitting region of an LED is small (e.g., in the range of 2 mm to 0.7 mm
across in many cases) which in theory opens up the possibility for highly
controlled distribution of light. However many of LED optics developed so
far do not produce controlled distributions, rather they typically
produce Gaussian like distributions which is the hallmark of somewhat
uncontrolled (random) light distribution, and is not ideal for most, if
not all applications.

[0003] A bare LED chip or an LED chip covered in an encapsulating
protective transparent hemisphere, emits light over an entire hemisphere
of solid angle, albeit with diminishing intensity at polar angles (zenith
angles) approaching π/2. FIG. 1 is a plot light intensity in arbitrary
units as a function of polar angle for a commercial high power LED.

[0004] FIG. 2 shows a reflector 202 arranged to collect a portion of light
emitted by an LED 204. A problem with using a reflector with an LED that
emits over the entire hemisphere of solid angle is that the reflector
needs to have an aperture and thus can not intercept and redirect all of
the light. As shown in FIG. 1 light emitted within polar angle range from
zero to φ passes through the aperture of the reflector 202 without
redirection or control. Additionally for the reflector 202 to exert
detailed control over the emitted light distribution it must be specular
as opposed to diffuse, and polishing a reflector sufficiently to make it
specular is often expensive.

[0005] In an attempt to address the problem posed by the hemispherical
range of light output from LED, a type of "primary" optic 302 shown in
FIG. 3 has been developed. (This is termed a "primary" optic because it
is assumed that it may be used in conjunction with a "secondary" optic
such as the reflector 202.) The term "primary optic" may also be taken to
mean an optic which has an optical medium of index >1 extending from
the LED die so that there only outer optical surfaces. The primary optic
302 is designed to intercept light emitted by an LED chip which is
positioned in a space 304 at the bottom of the primary optic 302 and to
redirect the light radially outward, perpendicular to an optical axis
306. The primary optic includes a refracting part 308 and a TIR (Total
Internal Reflection) part 310 both of which contribute to redirecting the
light. One drawback of the primary optic 302 is that because it includes
multiple optical surfaces that contribute to light in the same direction
it will increase the effective size of the source (also the etendue),
which reduces the controllability of light from the LED. The increased
effective size of the source can in some cases be compensated for, by
using larger secondary optics but this may be undesirable based on cost
and space constraints. By way of loose analogy to imaging optics, the
primary optic creates multiple "images" of the LED, e.g., one from the
refracting part 308 and one from the TIR part 310.

[0006] Although, the primary optic 302 is intended to redirect light
perpendicular to the optical axis, in practice light is redirected to a
range of angles. This is because the primary optic is small and
positioned in close proximity to the LED, and consequently the LED
subtends a not-insignificant solid angle from each point of the primary
optic, and light received within this finite solid angle is refracted or
reflected into a commensurate solid angle. The result is shown in FIG. 4
which is a plot of light intensity vs. polar angle for an LED equipped
with the primary optic 302. Although this distribution of light shown in
FIG. 4 is not especially suited to any particular application, it is
intended to direct light into an angular range that can be intercepted by
a secondary optic e.g., reflector 202. The goal is not fully achieved in
that the angular distribution of light produced by the primary optic 302
covers a range that extends from zero polar angle and therefore all of
the light can not be intercepted by the reflector 202.

[0007] Another presently manufactured commercial optic 502 for LEDs is
shown in FIG. 5. In use, an LED (not shown) will be located in a bottom
recess 504. This optic 502 is one form of "secondary" optic. A LED with
or without the primary optic 302 attached can be used. If used the
primary optic will fit inside the bottom recess 504. The secondary optic
502 is made from optical grade acrylic (PMMA) and is completely
transparent with no reflective coatings. The optic 502 includes a TIR
(Total Internal Reflection) parabolic surface 506 which collects a first
portion of light emitted by the LED, and a convex lens surface 508 which
collects a remaining portion of the light. Both surfaces 506, 508 are
intended to collimate light. As might be expected in actuality the light
is distributed in a gaussian like angular distribution over a certain
angular range which is variously reported as 5 degrees and 10 degrees.
The former value may be a FWHM value, and the actual value will vary
depending on the exact LED that is used. This design is only useful for a
fairly narrow range of specialized applications that require a far-field
highly collimated LED spotlight. FIG. 6 shows an angular distribution of
light produced by this type of optic. As shown the angular distribution
is gaussian-like not uniform.

[0008] In order to get a broader angular distribution of light some form
of surface relief pattern can be added to a top surface 510 of the optic
502 which is planar as shown in FIG. 5. Alternatively, the surface relief
pattern can be formed on a "tertiary" optic that is attached to the top
surface 510. One type of surface relief pattern-concentric rings of
convolutions is shown in a plan view in FIG. 7 and in a broken-out
sectional elevation view in FIG. 8. Another type of surface relief
pattern--an array of lenslets is shown in a plan view in FIG. 9 and in a
broken-out sectional elevation view in FIG. 10. FIGS. 11 and 12 show
light intensity distributions produced by commercial optics that have the
same general design as shown in FIG. 5 but which have top surfaces with a
surface relief pattern to broaden the angular distribution. The
distribution shown in FIG. 11 is designated as having a 15 degree
half-angle pattern and that shown in FIG. 12 a 25 degree half angle
pattern.

[0009] In fact at 25 degrees such designs are coming up against a limit.
The limitation is explained as follows. Given that the optic 502 with a
flat surface 510 nearly collimates light to within a nominal 5 degree
half angle, it can be infered that light is incident at the surface 510
at about 5/n degrees, where n is the index of refraction of the optic,
which for the sake of the following can be considered zero i.e.,
collimated. In order to create broader distributions of light, some
relief pattern as discussed above is added to the top surface 510.
Portions of the relief pattern will be tilted relative to the light rays
incident from below, and will therefore refract light out at larger
angles than would the flat top surface 510. However, at about 25 degrees,
depending on how much light loss will be tolerated a limited is
reached--in particular the transmittance of the surface starts to decline
rapidly. In this connection it is to be noted that according to Snell's
law in order deflect light at a particular angle, say 25 degrees, the
angle of incidence on the surface must be considerably larger than 25
degrees. FIG. 13, includes a plot 1302 that represents transmittance
versus deflection angle when passing from a medium of index 1.6 (a
typical value for visible light optics) into air. The X-axis in FIG. 13
is in radians. At the polar angle of 0.44 which is approximately equal to
25 degrees, the transmittance curve 1302 is already into a decline. The
transmittance shown by plot 1302 is better than attained in practice
because, at least, it does not take into account reflection losses
experienced when light passes into the optic 502 through the bottom
recess 504 of the optic. This is evidenced by reports of 90% and 85%
efficiency for collimating versions of the optic as shown in FIG. 5, not
96% which is the starting value of plot 1302. In contrast, plot 1304
which applies to illumination lenses according to certain embodiments of
the invention accounts for losses at both of two lens surfaces.

[0010] FIG. 14 shows another type of optic 1400 that is useful for
illumination. This optic includes a saw tooth TIR section 1402 and a
central lens portion 1404. The optic 1400 can collect a full hemisphere
of emission from a source and forms an illumination pattern with a
half-angle divergence (polar angle) about 30 degrees. This lens is
disclosed in U.S. Pat. No. 5,577,492. For this type of optic there will
be some loss of light from the intended distribution at the corners of
the saw tooth pattern, which in practice may not be perfectly sharp due
to manufacturing limitations. Additionally, due to its complex shape the
cost of machining and polishing molds for injection molding is expected
to be high. Additionally the '492 patent does address controlling the
distribution of light within angular limits of the beams formed. The
optic 1400 is already broad relative to its height. If an attempt were
made to broaden the polar angle range of the illumination pattern, the
TIR surfaces 1404 would have to be angled at larger angles, making the
optic even broader-perhaps impractically broad

BRIEF DESCRIPTION OF THE FIGURES

[0011] The present invention will be described by way of exemplary
embodiments, but not limitations, illustrated in the accompanying
drawings in which like references denote similar elements, and in which:

[0016] FIG. 5 is a secondary optic for an LED that produces a somewhat
collimated light beam;

[0017] FIG. 6 is a plot of light intensity versus polar angle produced by
the secondary optic shown in FIG. 5;

[0018] FIG. 7 is a top view of a pattern of ring convolutions that are
added to a top surface of the secondary optic shown in FIG. 5 in order to
obtain a broader angular distribution of light;

[0019] FIG. 8 is a broken out sectional view of the pattern of ring
convolutions shown in FIG. 7;

[0020] FIG. 9 is a top view of an array of lenslets that are added to the
top surface of the secondary optic shown in FIG. 5 also in order to
obtain a broader angular distribution of light;

[0021] FIG. 10 is a broken out sectional view of the array of lenslets
shown in FIG. 9;

[0022] FIGS. 11-12 show broader light intensity versus polar angle
distributions of light that are obtained by adding light diffusing
features such as shown in FIGS. 7-10;

[0023]FIG. 13 is a graph including plots of transmittance verses
deflection angle for prior art illumination lenses and lenses according
to embodiments of the present invention;

[0024] FIG. 14 is an illumination lens that includes a saw tooth TIR
section in addition to a central lens portion;

[0025] FIG. 15 is a graph including an X-Z coordinate system and
generatrices (profiles) of two surfaces a lens according to an embodiment
of the invention;

[0026] FIG. 16 is a graph including the X-Z coordinate system and profiles
of a refined version of the lens shown in FIG. 15 that has a modified
positive draft portion of its outer surface to facilitate injection
molding and a modified corresponding portion of the inner surface to
compensate for the modified positive draft portion and maintain light
distribution control;

[0027] FIG. 17 is a graph including the X-Z coordinate system and profiles
of a lens for producing a more uniform light distribution than the bare
LED and that has a modified positive draft portion of its inner surface
to facilitate injection molding and a modified corresponding portion of
the outer surface to compensate for the modified positive draft portion
and maintain light distribution control;

[0028] FIG. 18 is a graph including the X-Z coordinate system and profiles
of a lens that has a central refracting portion and outer Total Internal
Reflecting wings that together distribute light in a controlled manner
within a relatively narrow range centered on the Z-axis;

[0029] FIG. 19 is a graph including the X-Z coordinate system and profiles
of a lens that has Total Internal Reflecting wings located at the top and
a surrounding refracting portion that together distribute light in a
controlled manner within a narrow angular near the "equator" of a
co-incident polar coordinate system;

[0030] FIG. 20 is plot showing light distributions produced by the lens
shown in FIG. 16 and a similar lens compared to an ideal target
distribution for uniformly illuminating a flat area (e.g., floor, wall,
ceiling);

[0031] FIG. 21 is graph including the X-Z coordinate system and profiles
of a lens similar to that shown in FIG. 15 but for producing a light
distribution with a ˜65 degree half angle width;

[0032] FIG. 22 is a plot showing light distributions produced by the lens
shown in FIG. 21 and a similar lens compared to an ideal target
distribution for uniformly illuminating a flat area (e.g., floor, wall,
ceiling);

[0033] FIG. 23 is graph including the X-Z coordinate system and profiles
of a lens similar to that shown in FIG. 15 but for producing a light
distribution with a ˜55 degree half angle width;

[0034] FIG. 24 is graph including the X-Z coordinate system and profiles
of a lens similar to that shown in FIG. 18 but for producing a light
distribution with a ˜25 degree half angle width;

[0035] FIG. 25 is graph including the X-Z coordinate system and profiles
of a lens similar to that shown in FIG. 18 but for producing a light
distribution with a ˜15 degree half angle width;

[0036] FIG. 26 is a plan view of an LED based fluorescent replacement
fixture that includes an array of the lenses according to an embodiment
of the invention;

[0037] FIG. 27 is plan view of a round recessed lighting fixture that uses
a several of the lenses according to an embodiment of the invention;

[0038] FIG. 28 shows a portion of an LED luminaire according to an
embodiment of the invention; and

[0039] FIG. 29 is a flowchart of a method of making lenses according to
embodiments of the invention.

DETAILED DESCRIPTION

[0040] FIG. 15 is a plot of half-profiles (generatrices) of a first
surface 1502 and a second surface 1504 of a lens 1506 according to an
embodiment of the invention. The plots are shown in a coordinate system
that includes an X-axis and a Z-axis. The surfaces 1502, 1504 are
surfaces of revolution about the Z-axis (optical axis). The surfaces
1502, 1504 are joined by an annular edge surface 1508. The surfaces 1502,
1504 bound a body of transparent material, e.g., glass, plastic,
silicone. The origin of the coordinate system corresponds to the location
of the light source (e.g., an LED). By loose analogy to imaging optics,
the origin of the X-Z coordinate system can be considered the one and
only focus of the lens 1506. A single ray 1510 is shown emitted from the
origin and refracted by the lens surfaces 1502, 1504. Various angles
phi1,phi2, phi3 as will be described below are shown.

[0041] According to embodiments of the invention illumination lenses have
a first surface 1502 and a second surface 1504 such as shown in FIG. 1
shaped according to the following coupled differential equations:

[0042] n2 is the index of refraction of the lens 1506 defined by the
equations; n1 is the index of refraction of the surrounding medium (e.g.,
of air) which usually equals 1; phi1 is the polar angular coordinate
(zenith angle) of the first lens surface; phi3 is the polar angle (zenith
angle) of an ideal ray (a ray emitted at the origin) that was initially
emitted at angle phi1 after the ray has left the second surface 1504 of
each lens defined by the equations (see FIG. 15) and is given by:

where, phi1_MIN and phi1_MAX are the lower and upper limits polar angle
limits respectively of light collected by each lens 1506 defined by the
equations; phi3_MIN and phi3_MAX are the lower and upper limits
respectively of a predetermined specified output light intensity
distribution for each lens 1506 defined by the equations; rad_in(phi1) is
the light intensity distribution of the light source (e.g., LED) for
which the lens 1506 is designed; and rad_out(phi3) is the predetermined
specified output light intensity distribution for each lens defined by
the equations; phi2 is a polar angular coordinate of the second lens
surface and is given by:

with initial conditions and r2_ini for r1(phi1) and r2(phi1) respectively

[0043] EQU. 1 is solved numerically for to obtain a value of phi3 for each
input value of phi1 and DE1 and DE2 are integrated numerically, e.g.,
using the Runge Kutta integrator.

[0044] If phi1_min=phi3_min=0, EQU. 3 will be undefined at phi1_min=0. In
this case, instead of using EQU. 3.0 one can use the values of phi3
obtained from from EQU. 1 at two closely spaced points (e.g., spaced by
0.001) to obtain a finite difference approximation to dphi3/dphi1.

[0045] The most economical way to make the lens 1506 (and other
embodiments described by the equations given above and below) is by
molding e.g. injection molding or glass molding. To simplify the
construction of the molds used to mold the lens 1506, the first lens
surface 1502 can be molded by the core of the mold and the second lens
surface 1504 by the cavity of the mold. (This may not be possible for all
versions of the lens 1506.) Best practice in injection molding is to have
a draft angle of one-half to a few degrees. The design of the part to be
molded and the mold can not be such that the solidified molding material
will be locked into the mold. Referring to FIG. 15 it is seen that at
location 1512 the second surface 1504 of the lens is vertical (parallel
to the Z-axis). Versions of the lens 1506 in which this condition occurs
could be injection molded with a mold that has a parting line at location
1512, but this would not be best, as the gate residual would have to be
cut from the lens surface. A solution to this problem is as follows. The
lens equations given above are integrated to an intermediate value of
phi1 (corresponding to a value of phi2) at which the second surface 1504
has a slope corresponding to a desired draft angle and then second
surface 1504 of the lens is extended downward at that draft angle. In
FIG. 16 the intermediate point which in this example corresponds to a
draft angle of two degrees is labeled with reference numeral 1602. The
extension of the second surface at the draft angle is labeled 1604. The
portion extended at the draft angle 1604 is frusto-conical. The portion
of the profile of the second surface 1504 that is replaced by the
extension 1604, is labeled 1606. The second surface need only be extended
down far enough to intersect a ray emitted at phi1_MAX, after that ray
has been refracted by the first surface 1502, but can be extended
further. For example, integrally molded mounting features such as
mounting flanges and stand off pins can be located below the surfaces
defined by the generatrices shown in the FIGs. Because the extension at
the draft angle 2104 differs from the replaced section 2106, light rays
would not, if nothing else were done, be directed out of the lens at the
correct angles. To resolve this, and correct the deflection of rays back
to what would be obtained in the original lens 1506, a portion of the
first lens surface 1502 that refracts light to the draft angle extension
2104 is redefined by the following equation.

[0046] (Note that in order to maintain consistency with the definitions of
phi1, phi2 and phi3 (see FIG. 15), positive values of phiD are measured
clockwise from the Z-axis, thus what is referred to as a positive draft
angle in the injection molding art, will be entered as a negative value
in DE3)

[0047] The redefined portion of the first lens surface 1502, defined by
DE3 is labeled 1608 in FIG. 16. The portion it replaces is labeled 1610.
If the first surface 1502 were not modified by using DE3 to defined the
portion 1608 light would be refracted by the draft angle extension 2104
to polar angles beyond phi3_max. (Note that some amount of light may be
directed beyond phi3_max due to lack of perfect clarity of the lens, and
the finite light source size, the latter factor being subject to
mitigation by increasing the size of the lens 1506 relative to the
source).

[0048] Using the draft angle extension 1604 of the second surface 1504 and
the redefined portion 1608 of the first surface 1502 slightly reduces the
transmittance of the lens but the amount of decrease is insignificant
because of the small percentage of light effected by the redefined
surfaces 2104, 2108, and because the changes in the angle of incidences
due to the redefinition is small.

[0049] Note that DE1, DE2 and DE3 are defined in the domain of phi1. In
order to find the value of phi1 at which to start the draft extension
2104 and switch to using DE3, the following equation is used:

[0050] To use EQU. 4 (with sub-expression defined by EQU. 2 and EQU. 5) a
selected draft angle (e.g., 1/2 to a few degrees) is entered for PhiD and
the EQU. 4 is solved numerically for phi1 using a root finding method.
Note that each evaluation of EQU. 4 will involve integrating DE1 and DE2
up to a value phi in order to determine r1, r2. Once a value of phi1
corresponding to phiD is found, a corresponding value of r1 for the
initial condition for DE3 can be found by integrating DE1 up to this
value of phi1. The value of phi1 found in this way is referred to herein
below as phi1 at phiD.

[0051] For each of several examples discussed herein a table of inputs to
the lens equations is given. The table for the lens represented in FIG.
16 is:

[0052] Note that although the initial conditions and dimensions shown in;
the FIGs. can be considered to be in arbitrary units (meaning that
scaling is possible), the values were selected with millimeter units in
mind. In fact, a prototype discussed below was made with these dimension
in millimeters. The second to last row in the tables determines the Phi1
value at which the initial conditions r1_ini and r2_ini are defined. The
choice of r1_ini and r2_ini is not critical. The difference between
r1_ini and r2_ini should be chosen to give a designed initial lens
thickness. Alternatively, r2_can be adjusted to give a certain lens
diameter. One caveat is that if r1_ini and r2_ini are chosen too close
the profiles given by the lens equations may cross-over which is
physically excluded. The solution to this problem is to choose r1_ini and
r2_ini further apart and reintegrate the lens equations. Also, a smaller
difference between r1_ini and r2_ini will lead to a faster mold cooling
time and therefore increased manufacturing productivity. Furthermore,
r1_ini must be large enough to accommodate the LED.

[0053] The lens shown in FIG. 16 collects light energy from a full
hemisphere of solid angle from an LED and distributes the light
substantially uniformly on an area of a plane (e.g., floor, ceiling or
wall). Additionally, the light is substantially confined to a cone of
polar angle (zenith angle) 45 degrees. This is a good polar angle limit
for flood lighting. Some luminares used for general lighting emit light
in even broader angular ranges. The lens can readily be adapted to emit
over larger angular ranges by adjusting phi3_max. Of course, uniform
illumination of an area of a plane can not be obtained without limits on
phi3_max because as phi3_max approaches Pi/2 the light energy requirement
for any finite illumination level goes to infinity.

[0054] In FIG. 16 a series of design rays 1602 are shown emanating from
the origin and traced through the lens 1506. (Only two are connected to
lead lines so as not to crowd the drawings). One ray which is not visible
is along the +Z axis. Another ray which is initially not visible is
emitted, along the +X axis and is then refracted at an angle by the lens.
These are all ideal rays emanating from the origin of the X-Z coordinate
system. The initial angles of these rays are not arbitrary, rather the
angles are selected to divide the light energy emitted by the light
source (e.g., LED) into equal energy portions. Doing so helps to
visualize how the lens redistributes energy.

[0055] For certain combinations of the bare LED light distribution rad_in
and the desired output light distribution rad_out, DE1 defines a profile
of the first surface of the lens that has a negative draft near its edge
(positive in the convention of the present description) leading, to an
"undercut" condition. Such a lens could not be molded using a straight
forward mold design because the inner surface would lock onto the core of
the mold as the lens material hardened. Such a lens might be made using a
more expensive method e.g., using a core made of a low temperature
meltable material that is melted out. An example where the first surface
negative draft condition occurs is in the case that rad_in is nearly
Lambertian as shown in FIG. 1 and rad_out is set equal to 1.0 in order to
more uniformly distribution the light from an LED. FIG. 17 illustrates
this example along with lens profile corrections that are discussed
below. FIG. 17 shows a generatrix 1702 of a lens inner surface given by
DE1 and a generatrix 1704 of the lenses outer surface given by DE2 in
combination with DE1. A lower portion 1706 of the generatrix 1702 of the
inner surface has a negative draft 1704. In order to address this problem
a portion of the inner lens surface 1702 starting from a point at which
the surface reaches a suitable draft angle (e.g., 1/2 to 5 degrees) is
replaced by a conical portion 1708 that continues at that draft angle. In
order to compensate for the change of the inner surface, a portion 1710
of the outer surface is redefined according to the following lens
equation:

where, n1, n2 phi1, phi3, phiD are as defined above; r2_d1 is the polar
radial coordinate of the redefined portion 1710 of the second lens
surface 1704; phi1_phiD is the value of phi1 at phiD on the first surface
defined by DE1; r1_switch is the polar radial coordinate of the point on
the first surface 1702 at which the switch is made to the conical portion
1708, i.e., r1(phi1_phiD)=r1_switch. Although DE4 is defined in the
domain of phi1, the polar angular coordinate phi2 of the redefined
portion 1710 is given by:

Cartesian coordinate of the redefined portion 1710 can be obtained from
r2_d1 and phi2_d1. In order to find the value of phi1 at which the inner
lens surface has an angle equal to a desired draft the following equation
is used:

phiD = φ 1 - π 2 - θ 1 EQU .
7 ##EQU00009##

where theta_1 is the angle of incidence of ideal rays on the first
surface and is given by:

[0056] As in the case of EQU. 4, EQU. 7 (with theta_1 defined by EQU. 8)
is used by plugging in a selected value for phiD (e.g., 1/2 to a few
degrees) and using a root finding method to find the value of phi1 that
balances EQU. 7. Table II below list information for the lens shown in
FIG. 17.

[0057] FIG. 17 illustrates the ability to change the distribution of light
within the angular bounds of light emitted by the source without changing
the bounds themselves. Note that Phi1_MIN=Phi3_MIN=0.0 and
Phi1_MAX=Phi3_MAX=1.57 (90 degrees). The version of the lens shown in
FIG. 17 makes the light output more uniform than the bare LED.

[0058] As shown by plot 1304 in FIG. 13 as the deflection angle increases
beyond a certain point the transmission of the lens drops off
precipitously. For certain lighting tasks a narrow distribution of light
is desirable. If a high collection efficiency is to be maintained by
keeping phi1_max at 90 degrees then a higher deflection angle is needed
in order to produce a narrower distribution of light. FIG. 18 shows
generatrices of a lens 1800 that includes a central portion 1802 that has
a first surface 1804 and a second surface 1806 defined by DE1 and DE2 and
also has a conical surface 1808, an exit surface 1810 and a Total
Internal Reflection (TIR) surface 1812 given by DE5 below. The TIR
surface defined by DE5 works in concert with the central portion 1802 to
continue the overall light intensity distribution specified by rad_out.

%4:=tan(phi_draft) cos(phi1_switch)-sin(phi1_switch) Where, n1, n2, phi1,
phi3 are as defined above; r2--w is the polar radial coordinate of
the TIR surface 1812; r1_switch is the polar radial coordinate of the top
of the conical surface 1808 (also in the case of FIG. 18 the point at
which the conical surface 1808 meets the first surface 1804 defined by
DE1.) phi1_switch is the polar angular coordinate of the top of the
conical surface 1808; phi_draft is the angle of the conical surface 1808
measured in the clockwise direction from the positive Z-axis; phi_exit is
the angle of the surface normal of the exit surface measured in the
clockwise direction from the positive Z-axis, with initial condition
r2--w_ini. The polar angular coordinate (zenith angle) of the TIR
surface 1812 is given by the following equation.

r1--w and phi2w together define the TIR surface 1812 in polar
coordinates. Cartesian coordinates can be obtained from them.

[0059] In embodiments such as shown in FIG. 18 phi_draft has a small
negative value to allow the lens 1800 to release from a mold. A more
negative phi_draft will tend to increase the size of the TIR surface
1812. On the other hand a more negative value of phi_exit tends to reduce
the size of the TIR surface. Both phi_draft and phi_exit should be
selected (using phi1_max, phi1_switch and phi3_max as points of
reference) to avoid large angles of incidence that would reduce light
transmission. Note that the exit surface 1810 can be raised slightly from
the top edge of the TIR surface 1812 in order to provide a peripheral
location for an injection molding gate. The portion of the lens 1800
between the conical surface 1808, the exit surface 1810 and the TIR
surface 1812 is referred to herein as the "TIR wings". Table III below
lists information for the lens shown in FIG. 18.

[0060] Whereas the TIR wings shown in FIG. 18 are useful in confining
light to smaller angular range around the Z axis, the TIR wings defined
by DE5 can also be used to confine light to a small angular range near
phi3=π/2 (near the "equator" of a the spherical coordinate system).
FIG. 19 shows a lens 1900 that does this. In order to use the
differential equations given above to define a lens having TIR wings at
the top as shown in FIG. 19 the differential equations are integrated to
the left of the Z-axis, i.e., with negative values of the phi variables.
Note rad_out and rad_in are generally assumed to be symmetric so using
negative phi values does not change these light distributions. FIG. 19
shows a generatrix of a first surface 1902 defined by DE1 and a
corresponding generatrix of a second surface 1904 defined by DE2 in
combination with DE1. The first surface 1902 has a negative draft meaning
that it would lock onto the core of a mold and could not be removed. To
resolve this, most of the first surface starting from point 1906 at which
phiD=-177.5 degrees (a positive draft angle of 2.5 degrees) was replaced
by a constant draft conical section 1908 and a corresponding portion of
the second surface 1904 was replaced by a second surface 1910 defined by
DE4. The TIR wings of lens, 1900 include a TIR reflecting surface 1912
defined by DE5, a constant draft surface 1914 and an exit surface 1916.
In this special case in which phi_draft=-90 degrees the constant draft
surface 1914 is planar, as opposed to conical. Table IV below gives
information about the lens shown in FIG. 19.

[0061] R2_switch is the initial condition for DE5 which in the case of the
lens represented in FIG. 19 was integrated starting at phi1_phiD.
R2_switch was the final value of r2 given by DE2 at phi1_phiD.

[0062] Lenses defined by the lens equations given above are able to
collect a full hemisphere of light emitted by an LED, and are able to
distribute the light in a controlled manner. At the same time surfaces of
the lens defined by these equations are shaped to reduce transmittance
losses. The examples described while providing a wide variety of light
distributions hardly loose any more light by reflection than would an
optical window at normal incidence. The calculated transmittances for the
lens examples described herein are negligibly different from the
transmittance for light passing perpendicularly through an optical
window. As illustrated above, many practical general illumination lenses
defined by the differential equations given above the calculated
transmittance is over 90%. The second curve 1304 in FIG. 13 gives
transmittance as a function of deflection angle for lenses defined by DE1
and DE2. The deflection angle is equal to (phi3-phi1). For normal
incidence the transmittance through both surfaces, based on an index of
1.497, is 92.2%%. A transmittance of 90% represents a high optical
luminaire efficiency compared to standard luminares. Also, reflected
light may eventually scatter out of the lens into the beam pattern. This
effect may be increased making the area under the lens surrounding the
LED reflective. The optical luminaire efficiency is defined as the
percentage of light emitted by a light source (e.g., LED) that is output
by an associated luminaire which in the present case includes the lenses
defined by the above differential equations.

[0063] There is another efficiency factor that is termed herein "pattern
efficiency" and is related to the percentage of light energy in an output
distribution of light that is in excess of a required light intensity.
Because the light distribution patterns produced by most luminaries
(e.g., flood lamps, downlights) is stronger in a central part of an
angular or spatial range that is intended to be illuminated, the total
power of the luminaire must be higher than it would have to be if the
pattern of illumination covered the angular or spatial range uniformly.
Because the predetermined light output distribution rad_out(phi3) can be
freely specified and achieved to a degree of fidelity illustrated below,
lenses according to the above equations can produce light intensity
distributions that avoid wastefully excessive central intensities. If a
uniform light intensity distribution as a function of phi3 is needed then
rad_out(phi3) is set equal to one in the above equations. If a flat area
such as the floor of a room, desk or counter surface, is to be
illuminated uniformly without wasteful excessive central intensity then
rad_out(phi3) can be set to:

rad_out ( φ 3 ) = 1 ( cos ( φ 3
) ) e EQU . 10 ##EQU00013##

where e is approximately equal to 3, e.g., 3.2, 3.5.

[0064] This distribution with e=3.0 is a theoretically known distribution
and is shown as a plot 2002 in FIG. 20 for a phi3 range from zero to 45.
This distribution is quiet the opposite of the usual luminaire
distribution which is peaked in the center. This distribution is lowest
in the center and increases as the polar angle phi3 increases. The
increase is about a factor of 2.8 at 45 degrees. More light is required
at high values of phi3, because there is more area per phi3 increment as
phi3 increases. Examples of lenses defined using the intensity
distribution specified by EQU. 10 are shown in FIG. 15, FIG. 16, FIG. 18,
FIG. 21, FIG. 23, FIG. 24 and FIG. 25. According to embodiments of the
invention a higher fidelity to the distribution shown in FIG. 16 which is
based on e=3 is achieved if e is slightly higher than 3 e.g. 3.2, 3.5.
This is believed to be due to the fact that the finite size of the LED
die causes a blurring effect (akin to an angular analog of a point spread
effect, or apodizing effect) which leads to lesser variation than
intended. This is compensated by increasing e in rad_out of the form
given by EQU. 10. The amount that e should be increased can be determined
by making a few prototype lenses using different values of e. For example
one can start with a value of e=3 which will probably produce an actual
rad_out distribution that is too weak a function, then one can try 3.5
and depending on whether the variation of the resulting distribution
function is too strong or too weak one can then use a lower or higher
value of e. The inventor has found that a few prototypes are sufficient
to achieve acceptable fidelity to the intended distribution.

[0065] In FIG. 20 the data points denoted by diamonds are based on
measurements of the actual rad_out light distribution produced by a
prototype lens where rad_out in the equations was as given by EQU. 10
with e=3.0--the theoretical value. The data points denoted by circles are
based on measurements of the actual rad_out light distribution of a
second prototype lens based on a value of e=3.5. Generatrices of the
second prototype are shown in FIG. 16 and information relating thereto is
given in Table I above. As shown the data points for the second protype
points follow the target function more closely. The measurements were
done using a white Luxeon III LED manufactured by Lumileds of San Jose
Calif. There was an observed asymmetry in the placement of the LED die of
the Luxeon III used for testing which may have led to the asymmetry of
the measured rad_out distributions shown in FIG. 20 and FIG. 22. Numerous
IES files of prior art luminaries were reviewed and none were found that
matched EQU. 10 with e=3.0 to the degree achieved with the second
prototype. FIG. 20 does not convey the striking visual impact that the
inventor observed when these lenses were first tested. Arranging a single
LED with one of the lenses on a table to illuminate the ceiling one sees
a large clear uniform disk of light about 10 ft (3.05 meters) in
diameter. It is strikingly unfamiliar even to a person familiar with a
variety of modem light fixtures.

[0066] A more general way of correcting rad_out based on discrepancies
between the intended rad_out function and the measured rad_out function
is to first make a lens with rad_out set equal to the target distribution
and measure the actual rad_out distribution achieved at a number of
points, e.g., 10 points. Then, the data points are normalized so that the
integrated light intensity represented by the data points is equal to the
integrated light intensity of the same 10 points of the target
distribution. Next point-by-point differences are computed between the
normalized measurements and the target rad_out function. These
differences are then added to the points of the target rad_out function
to obtain a corrected set of points of rad_out. A spline fit of the
corrected set of points is then used as rad_out and the differential
equations reintegrated and a new lens made based on the new integration.
(Alternatively rather that a spline fit an interpolating routine is used)
This procedure can be done recursively. If one is inclined to rely on ray
tracing then ray tracing can be used to evaluate each new lens rather
than making actual prototypes, however the inventor has relied on
prototyping.

[0067] If it is desired to avoid a sharp shadow at the edge of the
illuminated area rad_out(phi3 given by EQU. 10 can be multiplied by a
function that is constant over a substantial portion of the phi3 range,
say up to 0.8 times phi3_max, and then tapers down gradually (e.g.,
linearly). In some cases edge effects that occur at phi3_max even without
altering rad_out(phi3) may provide sufficient tapering of the light
pattern edge.

[0068] In practice there may be as much to be gained in terms of pattern
efficiency by using lenses according to the present invention as there is
to be gained in terms of optical luminaire efficiency (i.e., the
percentage of light generated in the luminaire that escapes the
luminaire).

[0069] Additionally the lenses defined by the lens equations given above
have smooth surfaces with a limited number of corners which means that
the issue of light loss at numerous corners is avoided. Additionally
having smooth surfaces with a limited number of corners, means that the
molds to make the lenses and consequently the lenses themselves can be
made more economically.

[0070] FIG. 21 shows generatrices of a lens 2100 defined by DE1, DE2, DE3
and designed to produce an output light distribution rad_out according to
equation 14 with e=3.0. A portion of the outer surface 2104 was replaced
by a constant draft 2106 section and a corresponding part 2108 of the
inner surface 2102 was redefined by DE3. An adjustment of e to 3.2 in the
equations defining the lens produced better fidelity to the intended
light distribution pattern. Information for the lens shown in FIG. 21 is
given in Table V.

[0071] Note that r2_ini was selected using a 1st order ODE shooting
method to obtain a lens diameter of 20.0 mm. In particular, after each
integration r2_init was scaled by 10.0 divided by 1/2 the lens diameter,
this being continued for a few iterations until the lens diameter was
within an acceptable tolerance of 20.0 mm.

[0072] FIG. 22 shows data obtained with prototype lenses and the above
mentioned Luxeon III LED. Diamond symbol data points are for a lens based
on the e of EQU. 10 set to 3.0 (the theoretical value) and circle data
points are for the lens 2100 for which e was set to 3.2. Note that high
fidelity to the intended light distribution was achieved. Such a wide
light distribution can for example be used for low bay lighting, or for
indirect uplighting from a chandelier or torchiere. A "half lens" based
on revolving the generatrix around only 180 degree could be used for
sconce that mounted at 6' (1.83 meters) produces more than the usual
accent lighting by illuminating a large e.g. 4' (1.22 meters) semicircle
on an 8' (2.4 meter) ceiling above the sconce. A small mirror could be
placed behind the lens to confine the light emitted by the LED to the
azimuthal range from 0 to 180 degrees.

[0073] FIG. 23 shows a lens 2300 that is similar to those shown in FIG. 15
and FIG. 22 but for which phi3_max is 55 and with rad_out defined by EQU.
10 having an exponent e=3.3. Table IV below gives information related to
lens 2300.

[0076] Note that both phi_exit and phi1_switch were decreased relative to
the lens shown in FIG. 18. In choosing these and other parameters one
goal is the minimize the overall size. Another goal is maintain high
overall transmittance.

[0078] According to embodiments described above EQU. 1 specifies a
monotonic increasing relation between phi3 and phi1, i.e., as phi1
increases so does phi3. According to alternative embodiments of the
invention rather than using EQU. 1 the following alternative is used:

[0079] According to this alternative phi3 is a decreasing function of
phi1. This alternative is generally not as good because it leads to
higher average ray deflections (phi3-phi1) and thus more surface
reflection losses. One possible use is in a lens that includes two or
more portions including at least one defined using EQU. 1 and at least
one defined using EQU. 11. For example a first portion of lens which
covers a phi1 range from zero to an intermediate value of phi1 which
bisects the light intensity output of the light source into two equal
portions can be defined using EQU. 11 and a second portion of lens which
covers a remaining phi1 range can be defined using EQU. 1. For both
portions phi3_min can be set to zero and phi3_max to 45 degrees. Within
both portions in the limit that phi1 approaches the intermediate value of
phi1, the output ray angle phi3 will approach zero. Thus, the junctures
between the surfaces at the intermediate angle can be continuous and
smooth.

[0080] Whereas lenses defined using EQU. 1 or EQU. 11 serve to control the
distribution of light flux (e.g., lumens per steradian), for some
applications it is desirable to control the relation between phi3 and
phi1 in a different way. In such cases rather than using EQU. 1 or EQU.
11 in integrating the lens equations one can use another relation, such
as for example.

[0081] EQU. 12

φ3=mφ1

[0082] where, m is a constant angular magnification factor.

[0083] FIG. 26 shows the X-Z coordinate system with generatrices of lens
defined using phi3 giving by EQU. 12 with m=0.2 and DE1 and DE2.

[0084] FIG. 27 is a plan view of an LED based fluorescent replacement
fixture 2702 that includes an array of the lenses 2704 (lead lines for
only three are shown to avoid crowding the figure) defined by the
differential equations given above. Each lens 2704 controls the light
from a single LED chip or from a group of LED chips that are arranged
close together, for example in a single LED package. The fixture also
includes a power supply (not shown) for converting line power to power
for the LEDs. The fixture may also include individual heat sinks (not
shown) for each LED or LED package or a common heat sink. Heat sinks may
be thermally coupled to a surface 2206 of the fixture in order to provide
a larger area for dissipating heat.

[0085] FIG. 28 is a plan, view of a round (e.g., recessed, pendant, PAR
replacement) lighting fixture 2802 that uses several of lenses 2804
defined by the differential equations given above (only three of which
are numbered to avoid crowding the figure). Note that the lenses 2804 may
or may not be recessed above the ceiling level. Recessed lighting
fixtures are typically made in six and four inch diameter sizes. As in
the preceding cases the fixture 2802 will also include a power supply not
shown and a heat sink (not shown).

[0086] FIG. 29 shows a portion of an LED luminaire 2902. The luminaire
2902 includes a packaged LED 2904 mounted on a heat sink 2906. A lens
2908 defined by DE1, DE2 and DE3 is also mounted on the heat sink 2906.
The lens 2908 is located around the LED 2904 with the LED located at the
focal point (X-Z coordinate system origin) of the lens 2908.
Alternatively, unpackaged LED chip could be used. Alternatively, a lens
defined in part by DE4 and/or DE5 could be used.

[0087] FIG. 30 is a flowchart of a method 3000 of making lenses according
embodiments of the present invention. In block 3002 the values of the
variables and functions, as are listed in the tables above, are entered
into a computer that is loaded with a differential equation integrator
such the Runge Kutta routine, for example. In block 3004 a choosen subset
of the differential equations DE1, DE2, DE3, DE4, DE5 are integrated to
obtain an integrated solution. The integrated solution may be output as a
series of points along each generatrix and optionally associated normal
vectors for each point.

[0088] In block 3006 data representing the integrated solution is input in
a Computer Aided Manufacturing (CAM) program and processed to generate
machine tool control code.

[0089] In block 3008 the machine tool control code is entered into a
Computer Numeric Control (CNC) machine tool used to machine tooling
(e.g., mold inserts) for manufacturing lenses according to the integrated
solutions. Although not shown in FIG. 30, the mold inserts will need to
be hand polished (e.g., with a series of diamond pastes) before being
used.

[0090] In block 3010 the tooling is used to manufacture lenses according
to the integrated solutions.

[0091] Because the surfaces of the lens have smooth surfaces with few
corners monotonic injection molding molds to make them can easily be
turned and polished. Thus one can easily and relatively inexpensively
(e.g., compared to the case of FIGS. 7,8, 9, 10, 14) make versions of the
lens for each model of LED based on its light intensity distribution
rad_in(phi1). Moreover, if a particularly useful LED exhibits significant
unit-to-unit variations in the light intensity distribution rad_in(phi1)
then the LEDs can be binned by light intensity distribution pattern and a
version of the lens 1506 made for each bin. However, generally it will be
sufficient to base rad_in(phi1) on an average of light intensity
distributions for a particular light source.

[0092] According to alternative embodiments rather than use surfaces
defined by sweeping the generatrices defined above through a full 360
degrees, the physical lenses are truncated. For example the physical lens
can be truncated at the X-Z plane, and a mirror positioned at the X-Z
plane. The mirror will form an image of the LED, reflect substantially
all the light into a 180 degree azimuthal range and the lens 1506 will
then redirect the light as described above but within a limited azimuthal
range.

[0093] In addition to general illumination the lenses described herein can
be used for backlighting LCD displays. In backlighting applications the
lenses taught herein can be combined with a collimating lens to collimate
light distributed by the lenses taught herein. The associated collimating
lens can be use smooth portions, fresnel portions and/or saw-tooth TIR
portions.

[0094] Alternatively, a surface relief pattern can be added to one or more
of the surfaces of the lens in order to provide a degree of diffusion, in
this case the large scale profile of the lens surfaces 1502, 1504 is
described by the equations given above, but there is a short scale, small
amplitude variation added to the lens surface profiles.

[0095] Although the preferred and other embodiments of the invention have
been illustrated and described, it will be apparent that the invention is
not so limited. Numerous modifications, changes, variations,
substitutions, and equivalents will occur to those of ordinary skill in
the art without departing from the spirit and scope of the present
invention as defined by the following claims.